{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:7ALMNGUYIXI3TNVHTCHGZHKO5U","short_pith_number":"pith:7ALMNGUY","schema_version":"1.0","canonical_sha256":"f816c69a9845d1b9b6a7988e6c9d4eed378b80fcbc59beb4781b7b65670fc28e","source":{"kind":"arxiv","id":"1708.09779","version":1},"attestation_state":"computed","paper":{"title":"Few Sequence Pairs Suffice: Representing All Rectangle Placements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jannik Silvanus, Jens Vygen","submitted_at":"2017-08-31T15:40:40Z","abstract_excerpt":"We consider representations of general non-overlapping placements of rectangles by spatial relations (west, south, east, north) of pairs of rectangles. We call a set of representations complete if it contains a representation of every placement of $n$ rectangles.\n  We prove a new upper bound of $\\mathcal{O}(\\frac{n!}{n^6} \\cdot (\\frac{11+5 \\sqrt 5}{2})^n)$ and a new lower bound of $\\Omega(\\frac{n!}{n^4} \\cdot (4 + 2 \\sqrt2)^n)$ on the minimum cardinality of complete sets of representations. A key concept in the proofs of these results are pattern-avoiding permutations.\n  The new upper bound di"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1708.09779","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-31T15:40:40Z","cross_cats_sorted":[],"title_canon_sha256":"8c096c0290c5048c289bb594dda47aa17419ac5eec72591cb9ec0b0d216982ea","abstract_canon_sha256":"5c00a644164c152128fd17d58eec5261b2231ff0c072e5f596d00b524795b9c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:15.106883Z","signature_b64":"LxAJBCrmEyHoFE2vlb8doM1AHi2fzjWR0APteVp6S7rNvcm0WZDn6nFpecu3le6soZkjRp5uj/V9LkAx6He0Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f816c69a9845d1b9b6a7988e6c9d4eed378b80fcbc59beb4781b7b65670fc28e","last_reissued_at":"2026-05-18T00:36:15.106200Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:15.106200Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Few Sequence Pairs Suffice: Representing All Rectangle Placements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jannik Silvanus, Jens Vygen","submitted_at":"2017-08-31T15:40:40Z","abstract_excerpt":"We consider representations of general non-overlapping placements of rectangles by spatial relations (west, south, east, north) of pairs of rectangles. We call a set of representations complete if it contains a representation of every placement of $n$ rectangles.\n  We prove a new upper bound of $\\mathcal{O}(\\frac{n!}{n^6} \\cdot (\\frac{11+5 \\sqrt 5}{2})^n)$ and a new lower bound of $\\Omega(\\frac{n!}{n^4} \\cdot (4 + 2 \\sqrt2)^n)$ on the minimum cardinality of complete sets of representations. A key concept in the proofs of these results are pattern-avoiding permutations.\n  The new upper bound di"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.09779","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1708.09779","created_at":"2026-05-18T00:36:15.106309+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.09779v1","created_at":"2026-05-18T00:36:15.106309+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.09779","created_at":"2026-05-18T00:36:15.106309+00:00"},{"alias_kind":"pith_short_12","alias_value":"7ALMNGUYIXI3","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_16","alias_value":"7ALMNGUYIXI3TNVH","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_8","alias_value":"7ALMNGUY","created_at":"2026-05-18T12:31:03.183658+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7ALMNGUYIXI3TNVHTCHGZHKO5U","json":"https://pith.science/pith/7ALMNGUYIXI3TNVHTCHGZHKO5U.json","graph_json":"https://pith.science/api/pith-number/7ALMNGUYIXI3TNVHTCHGZHKO5U/graph.json","events_json":"https://pith.science/api/pith-number/7ALMNGUYIXI3TNVHTCHGZHKO5U/events.json","paper":"https://pith.science/paper/7ALMNGUY"},"agent_actions":{"view_html":"https://pith.science/pith/7ALMNGUYIXI3TNVHTCHGZHKO5U","download_json":"https://pith.science/pith/7ALMNGUYIXI3TNVHTCHGZHKO5U.json","view_paper":"https://pith.science/paper/7ALMNGUY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.09779&json=true","fetch_graph":"https://pith.science/api/pith-number/7ALMNGUYIXI3TNVHTCHGZHKO5U/graph.json","fetch_events":"https://pith.science/api/pith-number/7ALMNGUYIXI3TNVHTCHGZHKO5U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7ALMNGUYIXI3TNVHTCHGZHKO5U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7ALMNGUYIXI3TNVHTCHGZHKO5U/action/storage_attestation","attest_author":"https://pith.science/pith/7ALMNGUYIXI3TNVHTCHGZHKO5U/action/author_attestation","sign_citation":"https://pith.science/pith/7ALMNGUYIXI3TNVHTCHGZHKO5U/action/citation_signature","submit_replication":"https://pith.science/pith/7ALMNGUYIXI3TNVHTCHGZHKO5U/action/replication_record"}},"created_at":"2026-05-18T00:36:15.106309+00:00","updated_at":"2026-05-18T00:36:15.106309+00:00"}